The Infinite. Part 5. Is there anything infinite?

[All parts may be found here.]
In part 4 I looked at some “large” stuff. Now let’s think a little about the possibility that infinite things may exist.

Depending on the physics or metaphysics you subscribe to, infinite or eternal may be reality. In the physical universe we may live behind an “event horizon.” It’s possible there is a secret universe hiding behind some information barrier (inflation).[1] Think of the universe as existing on the surface of a ballon. There is an horizon.

M31. One of my favorite galaxies. It's bigger than our island.

Someone is blowing up this ballon. As it gets larger, the surface stretches and points formerly close together become further and further apart. There isn’t a real “center” of the universe, but everything is getting further from everything else. Space itself is expanding. Another interesting thing is going on: the further stuff is from you, the faster it is moving away from you.[2] Space is expanding locally, not into some extra dimension and not by pushing out some “edge.”[3]
But is space infinite? It is difficult to tell. But the point is, it may be, and that makes it reasonable to think of the infinite as more than just a mental construct perhaps. The data we perceive and interpret is a brain-mental process. The models we use to order that data often involve extremal properties that suggest the infinite.

Possible Two Dimensional Analogues of the Universe. Bottom version has the good hand right now. Illustration from Wikipedia.

But there are other reasons to consider the infinite “real.” Much of Mormon theological thought considers “time” a property always attached to both God and man. From Joseph Smith’s time, Mormon thought has contained a thread that appeals to both an infinite past and an infinite future. While special relativity suggests that frame of reference here is vital, it is also essential that “time” exists wherever the material exists. Since Mormonism is clearly in the camp of embodiment for God and certainly for Christ (no temporary Augustinian body here). Personal time is a fact for the mind in Mormonism. You “existed” before the physical state as a material being (some sort of matter[4]). Moreover the claim that man and God are coequal in temporal extension that they share an infinite “past.” Further, there is good textual evidence for that existence to be beginningless. Not in the sense of classical theism as a way to reconcile some of the antipodes of Christology, but in some objective sense.[5]

Physically we deal with time in chunks. Seconds, hours, and so on. We divide up seconds in to small pieces to increase accuracy of various things. In science, we see time in a Newtonian way (I’m not referring non-relativistic physics here). It is a totally ordered “thing” with events occurring along a “time line.”[6] Is there a practical limit to that sort of thing? In the sense of measurement, yes.

Really not possible to be sure what's going on in a time interval this short.

The problem is, it is just not clear if motion is somehow “continuous.” The question leads us to consider what it means to subdivide time “forever.”

The infinite as a reality? Certainly possible and theologically credible, even necessary. So, next time we’ll learn to parse it a bit. Once we get a little skill with that, we can come back to some of the questions that are below the surface of our theological claims.

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[1] Interested in more detail? See Lineweaver and Davis, Scientific American March 2005. “Misconceptions about the Big Bang.” A more technical version is in this interesting paper in astro-physics archive. I’m avoiding the multiverse speculations here. M theory will have to wait.

[2] As kids, my cousin and I used to sit around the campfire late into the night discussing what the boundary of the universe might be like. Was it like a big rocky sphere? What’s “beyond” it? After all, we had W. W. Phelps to go on.

[3] Edwin Hubble (1889-1953) is famous for cementing the view of an expanding universe. Hubble was a pioneer in another sense: his study of nebulae showed that the Milky Way was merely one among many galaxies.

[4] I think this belief may be overblown and I’m hazy about it in terms of it’s historically authoritative basis.

[5] Questions of memory and things like “what were you doing all that time?” are interesting, but I won’t approach them now. It is pretty obvious too, that if you subscribe to exhaustive foreknowledge for God, then you subscribe to an infinite actual knowledge base. Analytic philosopher David Braine’s book The Reality of Time and the Existence of God (OUP, 1988) seeks to account for Aquinas’ claims in a more formal setting and offers an interesting approach to God. A useful read I think.

[6] There is popular language referring to “nonlinear” time. This may reference something meaningful, I don’t really know. But I do know that it tends to be an escape hatch for the uninitiated. That is a rabbit hole we shall *not* go down.

Right. And I admit to not being trained in this area so could be completely misreading things. But even if you can divide time mathmatically beyond Planck time, if it is in reality impossible to do so, wouldn’t that be considered discrete?

You can subdivide it all you want. You’ll just never be able to measure what happens during that time. There a practical sense in which one could claim that shorter time intervals don’t exist. But I don’t make that claim. I suppose it’s a little like the tree in the forest.

I think that the infinite is a ridiculous argument. Logic would tell us that theoretically something could indeed be infinite, but in fact, reality shows us that only things which are countable exist. Sure, space could go on forever, but to show it is infinite is entirely impossible and thus a complete waste of time in pracitality. Think about it- infinity as a destination is entirely impossible, so is the possibility that two spots in an endless universe could be an infinite amount of space away from each other- impossible. What does this tell us about this supposed “unfinite”? Only that because it can never be known, it can’t be proved to exist. “Infinity” is a concept that allows us to think outside of a finite limit. In that sense it is better to use more practical words and terms such as something being “endless” or “without end”..

For instance- I can theoretically keep dividing the space of a square in half endlessly- I can really do that, as long as I keep doing it in eternity without ever stopping. But, I could never have an infinite count of dividing that space in half no matter how long I kept doing it- even if I never stopped. If I did it forever into eternity I could still never do it infinitely, only endlessly.

We read that unto God, all things are numbered. Logically then, He is not knowledgable of an endless amount of places in the universe. So, even if it did go on forever, all that even God can know are those places that have a spatial relation one to another and in order for this to be known there could never be two places that were not spatially numbered apart in some form of measurement. Thus by this test we can be assured that the universe is not infinite because something infinite has no meaning if it’s parts can be numbered individually thus defying its very definition.

I might also add- I will always only have a finite number of memories or answeres. I will only always have at any point, a finite knowledge of places and things. never will I have an infinite amount of memories, answeres, or knowledge of individual places in the universe, no matter how long I ever exist. So, even if I become a God, I will never have a capacity of knowing the infinite because it can’t exist.

Sure, space could go on forever, but to show it is infinite is entirely impossible and thus a complete waste of time in pracitality.

This is false. We need more precise measurements of density, but that is quite possible.

We read that unto God, all things are numbered. Logically then, He is not knowledgable of an endless amount of places in the universe. So, even if it did go on forever, all that even God can know are those places that have a spatial relation one to another and in order for this to be known there could never be two places that were not spatially numbered apart in some form of measurement.

As we will see in a later post, numbering something does not make it finite and that even in an infinite universe, any two locations are a finite distance apart.

If anyone is interested this technical point is explained in great detail in a set of YouTube lectures by Professor George Ellis. Specifically – “Cosmology, George Ellis, Lecture 1 part 3” at the 12 minute mark.

Rob,
I agree with you. Or at least I agree with the headache you gave me. Infinity is impossible to observe because we our(mortal)selves are finite. The best we can do is to observe that something is bigger or longer than us. Infinity we have to accept on faith.

One current hypothesis to explain the uniformity of the cosmic background radiation left over from the Big Bang at the start of our current observable universe includes a short period in which all matter was in close proximity and could transmit information to create uniformity, and then a period of very rapid inflation. This has been bolstered by the observations which won the Nobel Physics prize for Sol Perlmutter, et al, that the universal expansion is speeding up. There is a huge reservour of energy in “empty” space available to drive both phenomena. This has led to an additional hypothesis that the Big Bang may have been merely one instance of rapid inflation within an already existing universe, and that new daughter universes could bud off of an existing universe at any time. Since it would be creating its own space, the new universe would be outside of and unobservsble to our own.

This mechanism would be one that could create an unlimited succession of new daughter universes. What is more, it is consistent with such a succession having happened in the genealogy of our present universe. In other words, no single universe is infinite, but the succession of universes is unbounded in both time and space.

Recently a Mormon philosophy professor converted to Catholicism. In a blog post, he cited as one of his rationales the Catholic embrace of the Big Bang as a creation ex nihilo, contrary to Joseph Smith’s teachings about an eternal succession of worlds and gods. But the simple picture of the Big Bang as a transition from a state of no time or matter/energy into one of both dimension and substance, is no longer consideted valid. Stephen Hawking’s attack on the unmeasurable singularity at the time of the Bang is the point of his latest book and TV show. And the concept of an inflationary universe in which new big bangs are constantly repeating the process of creation matches Joseph Smith’s concept better than Augustine’s.

Other hypotheses offer up multi-dimensional “membranes” parallel in a higher dimension that occassionally collide and spawn a new Big Bang at the point of collision. This is not inconsistent with the recurrent inflation hypothesis.

When God tells Moses that the inhabited worlds he has created cannot be numbered to man, but are numbered to him, I read that as being a really high number, but not literally infinite. On the other hand, the argument Joseph made for the eternal nature of our spirits is that we had no beginning, and are thus not dependent for our existence on any temporary aspect of the universe.

I think we are mostly in agreement. Perhaps where we differ is in how we may view the infinite and how God would. I don’t think we are any different than the Gods themselves here concerning the concept of infinity. For instance- we may find that the universe can theoretically go on forever. So too can God. But, neither us nor God can actually know the infinite. So, if space really does go on forever, then God himself cannot possibly know the ends of it or “all” of it because that is impossible, even for a God, to know.

“For instance- I can theoretically keep dividing the space of a square in half endlessly- I can really do that, as long as I keep doing it in eternity without ever stopping. But, I could never have an infinite count of dividing that space in half no matter how long I kept doing it- even if I never stopped. If I did it forever into eternity I could still never do it infinitely, only endlessly.”

I disagree – if I never stop repeating an action then I will have completed that action more than a finite amount of times. In other words I will have completed the action an infinite amount of times (independent of if/when started completing the action).

Picture a number line that starts at zero and goes infinitely – the line is infinitely long even though it has a beginning.

“We read that unto God, all things are numbered. Logically then, He is not knowledgable of an endless amount of places in the universe. So, even if it did go on forever, all that even God can know are those places that have a spatial relation one to another and in order for this to be known there could never be two places that were not spatially numbered apart in some form of measurement. Thus by this test we can be assured that the universe is not infinite because something infinite has no meaning if it’s parts can be numbered individually thus defying its very definition.”

Again I disagree – just because a set can be numbered does not mean it is finite. The amount of numbers within the set of all integers is infinite – yet they are all numbered. It’s impossible to name an integer that does not have a unique name (i.e. number) assigned to it.

WVS-
Just gonna spew out thoughts again for your amusement:
1. The more I learn, the more I come to believe our universe really is discrete, not continuous. I think this implies that our sense of continuity, infinity, and associated mathematical constructs are approximations for what you said in your last article. I also think it’s a more parsimonious explanation, and one that is gaining traction the more we learn about both the very small, and very large. If that turns out to be true, does the theology change? That’s what really interests me.
2. Going along with #1 you said:

The infinite as a reality? Certainly possible and theologically credible, even necessary.

I’m not clear why it’s theologically credible, or necessary? Is it credible because of Joseph Smith words? Or because of what our ancient texts say? I would have a real problem with that argument given their highly fallible nature. In any case, if, as I said in #1 infinity is our approximation for something “really really big” could it not be that Joseph, or scripture was using that approximation due to its popular usage (as you also discussed in another article)? Also, why necessary? I find it much more palatable to believe in a God who had a god, that there’s an end to “exaltation” (seriously, can you imagine doing anything for forever. Think about it long and hard. At least for me, I can’t come up with ANYTHING I want to do FOREVER).
3. You said:

The problem is, it is just not clear if motion is somehow “continuous.” The question leads us to consider what it means to subdivide time “forever.”

Again, I think an approximation. I think Mach’s Bucket has to give up pause about a lot of these kinds of accepted theories. Indeed, I think Mach’s Bucket can be explained using knowledge we possess about forces between atoms, quarks, and other subatomic particles.
4. I must confess, spacetime and relativity are not my areas of expertise. Perhaps a bit more explanation would be helpful, for me at least. I get the gist, but couldn’t speak intelligently about it beyond a few Wikipedia articles.

You cannot complete something infinite- that is just the point. There would never be a time in your counting where you would have anything other than a finite count.

An infinite “set” is a contradiction or a paradox in and of itself. A set is something “complete” or “known”. Anything “infinite” is outside of what can be called “complete” or “known”. Dilbert’s hotel paradox is a musing in the stupidity that deal with just that kind of irrational thought.

jmb275, certainly room for your idea. I tried to explore something like that awhile back. But, if we take JS as normative for our beliefs (not necessary), then it leads to some discussion of the infinite.

Rob, the question of describing the infinite is important, but the idea that a “set is something complete or known” is just not true. And infinite sets are not paradoxical in and of themselves. Indeed, it’s quite possible that we live in one.

We read that unto God, all things are numbered. Logically then, He is not knowledgable of an endless amount of places in the universe. So, even if it did go on forever, all that even God can know are those places that have a spatial relation one to another and in order for this to be known there could never be two places that were not spatially numbered apart in some form of measurement. Thus by this test we can be assured that the universe is not infinite because something infinite has no meaning if it’s parts can be numbered individually thus defying its very definition.

Oddly enough, apparently Planck used dimensional analysis to arrive at a distance from fundamental constants. I am not sure what that means. Apparently no one really knows, either.

Why did he not use the charge on the electron as one of the constants, or unit spin or multiply by the fine structure constant to the 10th power? I am sure that a constant of length could be arranged from these additional universal constants.

It appears that the blackness of space has no end. It appears that time has no end. And what is countable to God is surely approximately infinite to me. This last is certainly hyperbole, that what is uncountable to man is countable to God.

And, as with advancing age, certainly infinite time can be crossed by finite memory. Every so often one wakes up with a spotless mind and exclaims, how wonderful to be alive, and who are you now? I am not worried.

My picture is that there is a closet in God’s house where all the 4 dimensional universes are kept, each one infinite in its dimension, but taking only a bit of 10 dimensional space. If a God were to wander into that closet by accident, we would see a strange projection.

Anyway, quantum mechanics seems to be necessary for life. (But, who can tell.) There are an infinite number of vacuum energy particles in an infinite space, none defined exactly in position or momentum or time and energy. If I cannot tell where something is by uncertainty, why would I worry about a distance much smaller than that uncertainty?

Off topic, but Joseph Smith used his ring to illustrate the eternal nature of spirits, saying that if something has a beginning, it will have an end. But then I wonder, what about sealings? Or the resurrection?

Fair enough. However, to be certain one mustbe able to prove that we live in an infinite “set” of the which is impossible to know. So, whereas we can talk about the nature of being part of an infinite set, so to speak, we shall never know it and it will always be something outside of the known and as such we can truly only be a part of that which is known and even though that number may be ever increasing forever, it will always be known by it’s individual finite parts and their absolute relation, spatially speaking and thus be connected in some finite value. In fact, nothing about this supposed “infinite set” will ever have any value beyond a finite placement one with another.

“Infinite” also connotes the inability of rational means to deal with the irrational. The golden ratio can never be absolutely pin-pointed by rational numbers, for example. One can calculate the Fibonacci series forever and never quite hit the nail on the head. However, by means of geometry it is easily generated and comprehended. Likewise, when we gaze into the night sky and wonder at the magnitude of the universe and our inability to comprehend it we should not be too surprised at the idea that God may have his own “geometry” of sorts — a medium for measuring that is yet unknown to us — that can easily bridle the infinite (or what may seem infinite to us at present).

If God could comprehend an infinite amount of things, there couldn’t possibly be room for any new ones to enter. By definition an infinite number is one that would be never ending and thus not a number at all. It stands to reason that even the Gods continue to have an increase in the number of things they know- ever growing yet never ending in their potential for things to know and yet also at the same time always forever away from all that is or can be forever known.

Let’s imagine that whole numbers may be counted ad infinitum. Now let’s add an infinite decimal count to every number. We have just sliced through eternity in a different direction even though, conceptually, the overall “count” of whole numbers has not increased.

Or, let’s imagine a circle as a symbol of infinity — no beginning or end. Now let’s imagine a sphere whereon an infinite number of circles may be found. Now let’s increase the size of the sphere! Or encapsulate it within a larger sphere, and another, and so on.

I guess I kind of agree with Rob in all of this. The only way we can really talk about infinity is in an abstract, almost Platonic sense (like perfect circles and triangles, etc.). Thought experiments dealing with infinity start off doing something mundane like counting or subdividing little discrete bits of something, and then they all ask you to make this leap right to doing it for infinite time. It’s like being on one ladder and climbing to another – at some point you have to shift your center of gravity and put your weight on the second ladder. Moving from actual discrete counting, measuring, etc., and then jumping to infinity seems like a jump from reality to abstraction. I think a more useful distinction is between the actually infinite and the potentially infinite – the latter being something that is being added through successive addition and could in theory continue for eternity, but could not of course ever “reach” there. I also think that, without a fixed temporal beginning, I have no idea why it is “now” and not some other time. “Nowness” seems quite puzzling without a beginning and then successive addition of time units between then and now. I am looking forward to the rest of this series because I think those arguing for an actual concrete infinity of anything have a strong burden to carry.

I don’t think an intelligent discussion of infinity can be had without some conceptional understanding of calculus, which I have entirely forgotten since taking it in 1975. But it seemed that Sir Isaac Newton figured out a few neat little math tricks to deal with degrees of infinity in a limited sense. The idea of an integral was to divide up the area under a curve into an infinite number of rectangles of infinitely small width and a given height and then add them up to get a finite and accurate number representing the area under the curve. Mathematics has tools that allow some of us to approach a few aspects of the infinite.

I think modern physics has investigated the infinite. One result is when Einstein sees time and space as the same thing, called space-time. Boggles my mind but might be a crucial concept when thinking about infinite time. I think quantum mechanics says that matter can only be in certain energy states, not any or all possible states, But general relativity with its wave-particles seems contrary to this. Why is the speed of light fixed and nothing can go faster? It seems like a modern version of sailing off the edge of the earth. Certainly the God of Mormonism with his physical body can travel faster than that. Or can he? Gravity I am told does not actually exist; it is is a result of large masses creating curves in space-time. So when the earth goes around the sun, it is actually going straight. (Useful to the alcoholic or sexual sinner when they deviate from the straight and narrow). Large enough masses like severe enough sins make black holes from which nothing can escape. Advanced physics is far more bizarre and counter-intuitive than even the most esoteric of the philosophies.

It raises the question: are our minds are even capable of understanding the way things really are? I like the analogy of the golden retriever patiently listening to his master read Shakespeare translated into Chinese and thinking about that inevitable doggie treat soon to come.

Jack,
Pretty good but here is another thought experiment of my own dealing with this infinity problem-

Suppose you had a box of finite size- perfectly square. Now lets say you were to add balls inside of that box and each ball had weight to it in relation to its size. The rules were that you could keep adding balls of ever decreasing size to fill all the voids. Disregarding the laws of physics even- you could go smaller than atoms to keep filling these voids. Potentially speaking- you could add an “infinite” amount of balls into the box of ever decreasing size. So does it also stand to reason then that you could then also be potentially adding an infinite amount of weight to the box?

This thought experiment shows very easily and clearly the problems of mixing reality with the imaginative. Of course you could never exceed the weight potential that the box was able to fill volume wise. The first ball filling the first void woud would always weigh more than all of the balls put in afterwards- even if you could put an infinite amount of them of all varrying weight.

Taking your sphere for example- sure, there may be an endless supply of points on that sphere, but the moment you add the concept of “how many” as a endpoint potential versus a sphere of different size you fail. More or less its like saying there are more infinite spots on a bigger sphere than on a smaller sphere. But I strongly disagree and I will show why this assumption is faulty-

Using Einsteins theory of relativity, we are going to disprove this in a sense of thinking in the Einstein way of things. Suppose you had two separate spheres- on in one galaxy and one in another. Now lets say that you had two spacemen who were by each sphere and without any other means of communication other than their magical telephone they were going to find out which sphere had more infinite spots over the other. Theoretically speaking, one would endlessly give exact and precise coodinates on his sphere infinitely through time until at last one would run out of spots of coordinates and thus the victor….right? This is where relativity comes into play to disprove this theory. Regardless of size, every coordinate on the bigger sphere will always match up exactly with the smaller sphere. But one might say- “but there are more potential spots on the bigger”. This is false and is thus proved by actually carrying the process out asoutlined above. Regardless of the size of a sphere- be it big or small, there are always the same amount of exact coordinates to spatially relate one point with another- that is all we can measure- the relativity of spots to each other.

We are not measuring relative size with these abstracts- that must be remembered.

“If God could comprehend an infinite amount of things, there couldn’t possibly be room for any new ones to enter. By definition an infinite number is one that would be never ending and thus not a number at all.”

Nobody can know an infinite amount of things. All thoughts, knowledge and memories will always be countable and finite if one were to stop and count them at any given moment. Even though we may have the capacity to increase that knowledge without end this does not mean that we will ever get to a point in the future that is infinitly far away from where we now are..

Rob — there are aspects of physical measurements which do require infinity. Mostly these are quantum mechanical effects, but they can easily be seen in any undergrad physics lab. One which has been experimentally performed is measuring the blackbody spectrum of the Cosmic Microwave Background. Deriving the blackbody spectrum requires summations (or integrals) of an infinite series. This spectrum matches the observations spectacularly. I wouldn’t be so harsh against the idea that infinity has no real impact on anything. Nature involves infinities quite often.

Physics seems to suggest that only a finite amount of information can be contained in a finite amount of space. So if our bodies’ (and God’s body) contain our memories and information can our bodies house an infitine amount of memories/knowledge?

Needless to say I’m excited to see what’s to come in your future posts!